Exercise 10, § 3

Let D be the group of displacements of $\mathbf{R}^3$ with determinant 1, the semi-direct product of $SO(3, \mathbf{R})$ and the group T of translations. We employ the notation $\sigma(\varphi, \psi, \theta)$ of Exer. 9, and denote by $t(\xi, \eta, \zeta)$ the translation with vector $(\xi, \eta, \zeta)$.

a) Let H be the closed subgroup of D that leaves $\mathbf{R}e_3$ stable and preserves orientation on $\mathbf{R}e_3$. Show that H is the product of the group of translations $t(0, 0, \zeta)$ and the group of rotations $\sigma(\omega, 0, 0)$. From this, deduce that the homogeneous space $E = D/H$, which may be identified with the space of oriented affine lines of $\mathbf{R}^3$, possesses a measure invariant under D, and, up to a constant factor, only one. (Make use of Exer. 5 a).)

b) Let $E_1$ be the open subspace of E formed by the oriented lines not orthogonal to $e_3$; such an oriented line can be determined by the coordinates $\xi', \eta'$ of its point of intersection with $\mathbf{Re}_1 + \mathbf{Re}_2$, and the coordinates $(\sin \varphi \sin \theta, -\cos \varphi \sin \theta, \cos \theta)$ of a direction vector. Show that a measure on $E_1$ invariant under $D$ is given by